How to twist radial vector to make spirals?
In order to produce a spiral effect, you need to effectively rotate the 'banding' based on the distance from the centre, 'wrapping' around at 1.0 back to 0.0 to continue the pattern.

This can be achieved as follows :



Note the lower branch of nodes - the Dot Product, Power, Multiply nodes. The Dot Product with both inputs set the same will generate the square of the distance from the origin. The Power(0.5) node takes the square root, resulting in the actual distance from the centre. The Multiply node can be adjusted to control the "tightness" of the turns of the spiral.

The Add node then adds this distance into the offset around the spiral - effectively rotating further around the centre the further away from the centre, while the Modulo(1.0) function allows the values to wrap around from 1.0 back to 0.0 to continue the pattern.

Note that I've also added an Add(1.0) and second Modulo(1.0) node - this is to allow for negative rotations to allow the spiral to be twisted in either direction (the normal Modulo function produces odd results when presented with negative values - the additional nodes correct for this - see https://blender.stackexchange.com/a/122489/29586 for more of an explanation for this).

I've also added a Multiply after the radial Texture node. This allows the number of repetitions of the banding to be adjusted. Stick with whole numbers to avoid breaking the pattern - eg, 3.0 will produce 3-times the number of bands.